Means: Many Groups (ANOVA)
Compute power or required sample size for a planned one-degree-of-freedom contrast in a one-way ANOVA with equal within-group variances. You can enter a custom contrast or generate all pairwise mean comparisons. The calculation uses the noncentral F distribution with denominator degrees of freedom N − G.
This calculator follows the equal-variance one-way ANOVA contrast procedure documented in the statistical literature/this method. A contrast is a linear combination of the group means, C = Σi ciμi, with coefficients that sum to zero. A pairwise comparison is a special case, such as (−1, 1, 0, …), which estimates the difference between two group means.
For equal group sizes n and a common within-group standard deviation σ, the standard error is
The alternative contrast is δ1 = Σ ciμ1i. The corresponding noncentral t parameter is Δ = δ1 / σC; the one-degree-of-freedom noncentral F calculation uses Δ2. With G groups and total sample size N, the denominator degrees of freedom are N − G. This is computed as two-sided power as the probability of exceeding the central F critical value on the corresponding noncentral-F distribution.
When solving for sample size, the calculator searches over integer subjects per group and returns the smallest equal per-group sample size whose computed power is at least the requested target. This mirrors the binary-search approach to sample-size determination while retaining integer group sizes.
Selecting pairwise comparisons creates every pair of groups using a coefficient vector with −1 for the first group, +1 for the second, and 0 elsewhere. The resulting tests use the same common within-group variance estimate and N − G denominator degrees of freedom. Because several hypotheses may be tested, the displayed pairwise powers are individual-test powers at the specified alpha; this calculator does not apply a Bonferroni, Holm, Tukey, Dunnett, or other family-wise multiplicity adjustment.
The this method procedure assumes continuous outcomes, normally distributed observations within groups, equal within-group variances, independent groups, and random sampling. The contrast coefficients must sum to zero. These assumptions are important because the reported significance levels and power depend on the one-way ANOVA model.
this method Chapter 539 Example 2 uses α = 0.05, G = 3, n = 5 per group, means {1, 2, 3}, contrast {−2, 1, 1}, and σ = 5. This yields power = 0.0797 for this configuration. The implementation reproduces power = 0.07972.
the software, LLC. this method Sample Size Software: One-Way Analysis of Variance Contrasts Assuming Equal Variances, Chapter 539. the relevant methodological literature, current procedure documentation. The chapter identifies the method, assumptions, noncentral-F power calculation, examples, and validation calculations.
Desu, M. M., & Raghavarao, D. (1990). Sample Size Methodology. Academic Press.
Fleiss, J. L. (1986). The Design and Analysis of Clinical Experiments. John Wiley & Sons.
Kirk, R. E. (1982). Experimental Design: Procedures for the Behavioral Sciences. Brooks/Cole.